Logically, this device should have existed somewhen in
History, but I've never heard of it. Sure, I know that a
"sling" is a type of rock-thrower. However, a sling does
not resemble an atlatl (see link) in the slightest. So,
why
isn't there a rock-thrower that resembles an atlatl?

I have
a suspicion as to the Answer of that Question,
and
it relates to the two notions of "hold" and "release".
When
you put the back end of a spear into an atlatl, and use
your
arm to accelerate the device, there is a "lever arm"
effect
that enables the spear to become separated from the
atlatl
in a good way (good for making the spear travel a longer
distance than if the atlatl was not used).

A rock, however, cannot exhibit any such lever-arm
effect.
When it is placed into the holding-part of our atlatl-
equivalent, such that we can accelerate it, there is
nothing
in the overall natural motion to cause the rock to escape
that holding-part. So the rock stays held right through
the
point where we might want to release it, for a long-
distance throw.

Meanwhile, the different design of a sling easily
accommodates both "hold" and "release", because the
user
simply lets go of one of the cords at the proper moment.
This lets the rock escape the holder-part and begin
moving
tangentially away from what had been the acceleration-
circle. (By the way, old carvings of armies that fielded
both archers and slingers had the slingers Behind the
archers, because they could out-range the archers.)

So, to have an atlatl that works for rocks, we need to
add
something New, a way to make it release the rock at the
proper point. What I shall propose, just to have a
completed Idea, involves a kind of "door"... (although
plenty of other methods will probably work as well).

Imagine the holder-part of our Rock Thrower to be
shaped
somewhat like a box. It has a bottom and four sides,
but
no top (because we put the rock into the box through
that
opening). When the Rock Thrower is accelerated, the
centrifugal effect causes the rock to impart force
against
one of the sides of its box. THAT is the side we convert
into a door.

Not exactly a door, however; think "see-saw", which has
a
pivot-point and two sections that move in relation to the
pivot. We want one side of the see-saw to be our door,
and we want the other side to be a kind of control-point.
Now we add, alongside the main shaft of our atlatl-
equivalent, a simple lightweight hollow rod that presses
against the above control-point. The hand holding the
Rock Thrower uses its thumb to press against the rod,
which pushes against the control-point, which keeps the
"door" closed on the box holding the rock.

At the right moment during the acceleration of the Rock
Thrower, the thumb is moved. The centrifugal effect,
acting on the rock, easily causes it to overcome the
lightweight rod's own centrifugal effect upon the control
point (especially because we can ensure the rod touches
the control point very near the pivot; the "force times
distance" of the rock will greatly exceed the force-times-
distance of the rod). Thus the door is pushed open and
the rock
escapes, free to now travel a nice long distance.

As a perhaps-slightly-better alternative, the thumb
might be inside a Ring at the end of that hollow rod, and
we physically attach the rod to the control-point
(probably with a hinge). Moving the
thumb could mean pulling on the rod and pulling on the
control point, thus directly opening
the door, so that none of the rock's available Force need
be expended to escape. It could travel farther, that is.

[Flying Toaster] (and others who added some links),
to the best of my knowledge, in order to properly
use those devices, you have to suddenly stop moving
the device in order for the accelerated object to
escape. But that is not how you use an atlatl. And
that is not how you use the Invention here.

// you have to suddenly stop moving the device in order for the accelerated object to escape //

True for the tennis ball thrower. I think for a staff sling you mearly have to stop accelerating it. The sling portion will then catch up, and trigger the release.

It does seem like a device with a precise release control could be more accurate with less practice, than a staff sling, but that advantage might be outweighed by the fact that your thrower is less energy efficient. With a staff sling, the total radius of the rock's path is covered partly by the staff and partly by the sling, so the rigid portion of your thrower will need to be longer (and heavier) than an equivalent staff sling. The box on the end will also add mass. To get the same velociy for the same rock, you have to accelerate more mass to the same angular velocity. After release, that extra mass at the end of your stick is also a disadvantage if you want to avoid slamming it into the ground.

[lurch], let us see if an ASCII sketch can be useful
here:
|___________________
|========================
|__|

Due to the limitations of ASCII sketches here at
the HalfBakery, I'm drawing this with the handle
on the right and the "box" at left, only upside-
down. The double-lines represent the main body
of the device; the single long horizontal line at top
is the lightweight rod, ending where a thumb
could affect it. (I would have made them longer
but ran into the 30-character limit.)

The short horizontal line at the bottom-left of the
sketch is the Bottom Edge Of A Side-Wall of the
box, so the box is actually Open in the downward
direction; you put the rock through that opening
into the box.

At the left side of the sketch (three vertical lines
are actually One line) is the see-saw Door. When
the whole thing is held so it is vertical, the Door is
at the top of the device.

Normally, of course, the device starts out
horizontal with the box opening upward. Insert
rock, grab handle, push rod against the control
surface of the see-saw Door,
and start accelerating the device toward the
vertical orientation. At the appropriate moment
(near the vertical orientation moment)
move thumb so that the Door can open and let the
rock fly.

[spidermother], I did say in the main text that
there are probably a lot of ways to do what I've
described. I think I did know about "staff slings",
but discounted it as being more of a variant of the
sling rather than of the atlatl. Your opinion may
vary, of course.

Just going by the description and the pictures, the
sling seems to be short compared to the staff, acting
mainly to hold and release the stone (unlike a hand
sling or a trebuchet, where the sling is much longer).
I suppose it's a matter of degrees.

Sounds to me like all you really need is a modified golf
club, suitable for hitting rocks while they sit on the
ground. The main modification would be some
padding on the head of the club, to keep from
shattering the rock at impact.